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arXiv:1812.04915 · Physical Review A

Fluctuations and photon statistics in a quantum metamaterial near a superradiant transition

D. S. Shapiro, Alexey N. Rubtsov, S. V. Remizov, W. V. Pogosov, Yu. E. Lozovik

Published Jun 17, 2019• 5 citations
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Abstract

The analysis of single-mode photon fluctuations and their counting statistics at the superradiant phase transition is presented. The study concerns the equilibrium Dicke model in a regime where the Rabi frequency, related to a coupling of the photon mode with a finite-number qubit environment, plays the role of the transition's control parameter. We use the effective Matsubara action formalism based on the representation of Pauli operators as bilinear forms with complex and Majorana fermions. Then, we address fluctuations of superradiant order parameter and quasiparticles. The average photon number, the fluctuational Ginzburg-Levanyuk region of the phase transition, and Fano factor are evaluated. We determine the cumulant-generating function which describes a full counting statistics of equilibrium photon number. Exact numerical simulation of the superradiant transition demonstrates quantitative agreement with analytical calculations.

DOI
10.1103/physreva.99.063821

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The analysis of single-mode photon fluctuations and their counting statistics at the superradiant phase transition is presented. The study concerns the equilibrium Dicke model in a regime where the Rabi frequency, related to a coupling of the photon mode with a finite-number qubit environment, plays the role of the transition's control parameter. We use the effective Matsubara action formalism based on the representation of Pauli operators as bilinear forms with complex and Majorana fermions. Then, we address fluctuations of superradiant order parameter and quasiparticles. The average photon number, the fluctuational Ginzburg-Levanyuk region of the phase transition, and Fano factor are evaluated. We determine the cumulant-generating function which describes a full counting statistics of equilibrium photon number. Exact numerical simulation of the superradiant transition demonstrates quantitative agreement with analytical calculations.
In view of recent proposals for the realization of anisotropic light-matter interaction in such platforms as (i) nonstationary or inductively and capacitively coupled superconducting qubits, (ii) atoms in crossed fields, and (iii) semiconductor heterostructures with spin-orbital interaction, the concept of a generalized Dicke model, where coupling strengths of rotating wave and counter-rotating wave terms are unequal, has attracted great interest. For this model we study photon fluctuations in the critical region of normal-to-superradiant phase transition when both the temperatures and numbers of two-level systems are finite. In this case, the superradiant quantum phase transition is changed to a fluctuational region in the phase diagram that reveals two types of critical behaviors. These are regimes of Dicke model (with discrete Z2{\mathbb{Z}}_{2}Z2​ symmetry), and that of anti-Tavis-Cummings and Tavis-Cummings U(1)U(1)U(1) models. We show that squeezing parameters of photon condensate in these regimes show distinct temperature scalings. Besides, relative fluctuations of a photon number take universal values. We also find a temperature scale below which one approaches a zero-temperature quantum phase transition where quantum fluctuations dominate. Our effective theory is provided by a non-Goldstone functional for condensate mode and by Majorana representation of Pauli operators. We also discuss the Bethe ansatz solution for integrable U(1)U(1)U(1) limits.
We consider a driven single-mode Dicke-Hamiltonian coupled to a dissipative zero-temperature bath. We derive the cumulant generating function for emitted photons of this quantum-critical system by using a PPP representation of the master equation in the thermodynamic limit. This cumulant generating function is shown to consist of two parts: a macroscopic component, which is Poissonian in nature with characteristic rate proportional to the order parameter of the system, and a part describing fluctuations, which is nontrivial in form and divergent around the quantum phase transition.
The controllability of current quantum technologies allows one to implement spin-boson models where two-photon couplings are the dominating terms of light-matter interaction. In this case, when the coupling strength becomes comparable with the characteristic frequencies, a spectral collapse can take place, i.e., the discrete system spectrum can collapse into a continuous band. Here, we analyze the thermodynamic limit of the two-photon Dicke model, which describes the interaction of an ensemble of qubits with a single bosonic mode. We find that there exists a parameter regime where two-photon interactions induce a superradiant phase transition, before the spectral collapse occurs. Furthermore, we extend the mean-field analysis by considering second-order quantum fluctuation terms, in order to analyze the low-energy spectrum and compare the critical behavior with the one-photon case.
The existence of zero-dimensional superradiant quantum phase transitions seems inconsistent with conventional statistical physics. This work clarifies this apparent inconsistency. We demonstrate the corresponding effective field theories and finite-temperature properties of light-matter interacting systems, and show how this zero-dimensional quantum phase transition occurs. We first focus on the Rabi model, which is a minimum model that hosts a superradiant quantum phase transition. With the path-integral method, we derive the imaginary-time action of the photon degrees of freedom. We also define a dynamical critical exponent as the rescaling between the temperature and the photon frequency, and perform dimensional analysis to the effective action. The dynamical critical exponent shows that the effective theory of the Rabi model is a free scalar field, where a true second-order quantum phase transition emerges. These results are also verified by numerical simulations of imaginary-time correlation functions of the order parameter. Furthermore, we also generalize this method to the Dicke model. Our results make the zero-dimensional superradiant quantum phase transition compatible with conventional statistical physics, and pave the way to understand it in the perspective of effective field theories. Published by the American Physical Society 2024
In this paper, we investigate the finite-temperature properties and phase transition of the Dicke model. Converting the atomic pseudo-spin operator to the two-mode Fermi operators, we obtain the partition function in terms of the imaginary-time path integral. The atomic population and average photon number as analytic functions of the atom-photon coupling strength are found from the thermodynamic equilibrium equation, which leads to the stationary state at a finite temperature and is determined by the variation in an extremum-condition of the Euclidean action with respect to the bosonic field. In particular we study the phase transition from normal to superradiation phase at a fixed low-temperature, in which the phase transition is dominated by quantum fluctuations. The phase transition induced by the variation of the atom-photon coupling strength indeed obeys the Landau continuous phase-transition theory, in which the average photon-number can serve as an order parameter with non-zero value that characterizes the superradiation phase. In the zero temperature limit our results recover exactly all those obtained from the quantum phase transition theory at zero temperature. In addition, we discuss the thermodynamic properties and compare the difference between finite-temperature phase transition and zero-temperature quantum phase transition. It is discovered that the average photon-number and mean energy in the low-temperature stationary state coincide with the corresponding values of zero-temperature in the strong coupling region. The entropy of the superradiation phase decays rapidly to zero with the increase of coupling strength.
The Dicke model famously exhibits a phase transition to a superradiant phase with a macroscopic population of photons and is realized in multiple settings in open quantum systems. In this paper, we study a variant of the Dicke model where the cavity mode is lossy due to the coupling to a Markovian environment while the atomic mode is coupled to a colored bath. We analytically investigate this model by inspecting its low-frequency behavior via the Schwinger-Keldysh field theory and carefully examine the nature of the corresponding superradiant phase transition. Integrating out the fast modes, we can identify a simple effective theory allowing us to derive analytical expressions for various critical exponents including the dynamical exponent. We find excellent agreement with previous numerical results when the non-Markovian bath is at zero temperature; however, contrary to these studies, our low-frequency approach reveals that the same exponents govern the critical behavior when the colored bath is at finite temperature unless the chemical potential is zero. Furthermore, we show that the superradiant phase transition is classical in nature, while it is genuinely nonequilibrium. We derive a fractional Langevin equation and conjecture the associated fractional Fokker-Planck equation that captures the system's long-time memory as well as its nonequilibrium behavior. Finally, we consider finite-size effects at the phase transition and identify the finite-size scaling exponents, unlocking a rich behavior in both statics and dynamics of the photonic and atomic observables.